model reduction for compressible ﬂuid-solid coupling and...

Model reduction for compressible fluid-solid coupling andits application to blasting

D. Xiaoa,b, P.Yanga, F. Fanga,∗, J. Xianga, C.C. Paina, I.M. Navonc, M. Chend

aApplied Modelling and Computation Group,Department of Earth Science and Engineering,Imperial College London,

Prince Consort Road, London, SW7 2BP, UK.URL:http://amcg.ese.imperial.ac.ukbChina University of Geosciences, Wuhan, 430074, China

cDepartment of Scientific Computing, Florida State University,Tallahassee, FL, 32306-4120, USAdWuhan University, Wuhan, 430072, China

Abstract

This work presents the first non-intrusive reduced order model (NIROM) for com-pressible fluid and structure interactions. The NIROM is constructed through properorthogonal decomposition (POD) and a radial basis function(RBF) multi-dimensionalinterpolation method. The novelty of NIROM for blasting modelling lies in reduced or-der modelling for the unique combination of fluid, solid, air-solid coupling and fracturemodelling. In this work, NIROM is applied to fluid modelling where an intensive com-putational cost is required. A coupling source term is introduced to fluid modelling.A Mohr-Coulomb failure criterion with a tension cut-off is used to judge whether newfractures are generated.

The performance of the NIROM for a structure interacting with compressible fluidflows in the presence of fracture models is illustrated by twocomplex test cases: abending beam forced by flows and a blasting test case. The numerical simulation resultsshow that the NIROM is capable of capturing the details of compressible fluid andstructure interactions and fractures while the CPU time is reduced by several orders ofmagnitude. In addition, the issue of whether or not to subtract the mean before applyingPOD is discussed in this paper. It is shown that solutions of NIROM without meansubtraction before performing POD are better than that NIROM with mean subtraction.

Keywords: ROM, compressible fluid-structure interaction, blasting

1. Introduction

The numerical simulation of fluid/solid interaction has attracted much attention ina wide variety of research areas. This problem is of significance to many fields inengineering such as aerospace engineering, biomedical engineering, wind turbines andblasting. However, the computational expense involved in solving such problems is so

∗Corresponding authorEmail address:[email protected] (F. Fang)

Preprint submitted to Elsevier December 24, 2015

great that this has hindered development in these areas. In order to address the issue ofhigh computational expense, this paper proposes a non-intrusive reduced order modelto solve fluid/solid interaction problems in an efficient manor.

Reduced order modelling is a technique that is capable of reducing the dimension-ality of large systems, thus resulting in a considerable increase in computational effi-ciency. POD is the method most widely used to form reduced order models and it aimsto represent a large system with only a relatively small number of optimal basis func-tions. The POD has been used successfully to various areas such as air pollution [1],ocean modelling [2, 3], fluid mechanics [4, 5, 6], aerospace design [7], neutron pho-ton transport [8], porous media [9, 10] and shape optimization [11]. The large systemcan be projected onto a reduced system by Galerkin projection to derive the ROMs.However, the use of POD/Galerkin raises numerical instability and non-linearity in-efficiency problems [12, 13]. Several methods have been presented to improve thenumerical stability of ROM such as calibration [14, 15], Fourier expansion [16], regu-larisation [17] and Petrov-Galerkin method [2, 18]. In order to enhance the non-linearefficiency, various methods also have been proposed, includingempirical interpola-tion method (EIM) [19], the discrete version of EIM (DEIM) [13], quadratic expan-sion method [20, 21], a hybrid of DEIM and quadratic expansion (residual DEIM)method [22], Petrov−Galerkin projection method [14], and Gauss−Newton with ap-proximated tensors (GNAT) method [23]. To the best of our knowledge, althoughthere are some works addressing reduced order modelling of fluid-structure interactionproblem [24, 25, 26, 27, 28, 29], constructing ROM for compressible fluid and struc-ture interaction problem in a non-intrusive way has not beenreported in the literature.In addition, this paper is the first time to numerically simulate the blast test case using anon-intrusive reduced order model. This is achieved by using a set of RBF interpolationfunctions(hypersurface) to represent the POD coefficients in reduced space.

The non-intrusive reduced order modelling technology is proposed to tackle thedisadvantages of the intrusive ROM such as dependence of thegoverning equationsand difficulties in modifying the source code. The modifications of source code couldbe impossible in commercial software [30]. A number of NIROM methods have beenproposed. such as a black-box stencil interpolation method[30], a POD-RBF methodfor unsteady fluid flows [31], a Taylor series and Smolyak sparse grid method for theNavier-Stokes equations [32], a two-level NIROM based on POD-RBF method fornonlinear parametrized PDEs [33, 34], a POD-RBF for the Navier-Stokes equations[35]. NIROM has also been applied to realistic problems such as 3D free surfaceproblem [3] and multi-phase flow in porous media problems [10].

This article, for the first time, applies the non-intrusive reduced order modellingmethod to compressible fluid and structure interaction problems, especially the highlynon-linear problem - the blasting test case. This model has been implemented underthe framework of a combined finite-discrete element method based solid model(Y2D)and an unstructured mesh finite element model (Fluidity).

In this approach, the results of high fidelity full model are recorded using snapshotsmethod, then a number of POD basis functions are generated from those snapshotsthrough singular value decomposition (SVD) method. A set ofhypersurfaces is thenconstructed to represent the compressible fluid and structure interactions and fracturemodel using the RBF-POD method. After obtaining the hypersurface, the reduced

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system of the compressible fluid and structure interactionsand fracture model is solvedby inputting earlier time levels’ POD coefficients into the hypersurface.

During the POD process, the mean of the snapshots is normallysubtracted. Theproblem of mean subtraction was discussed in [36, 37]. In their work, there was notmuch difference between the results with mean subtraction and results without meansubtraction. In this paper, the solutions with mean subtraction and without mean sub-traction are presented and discussed.

The performance of this compressible FSI NIROM without and with mean sub-traction has been assessed for two test cases: a bending beamforced by flows and ablasting test case. Comparison between high fidelity full model and the compressibleFSI NIROM without mean subtraction using different number of POD bases has beenmade to validate the accuracy.

The structure of the paper is arranged as follows: section2 presents the compress-ible fluid and solid coupling equations; section3 derives the formulation of a non-intrusive reduced order model for compressible fluid and structure interactions andfracture problems using POD-RBF method; section4 demonstrates the capability ofthe derived methodology by two numerical examples: a bending beam forced by flowsand a blasting test case. Finally in section5, the conclusion is drawn.

2. Description of compressible fluid/solid coupling and fracture modelling prob-lem

This work is carried out under the framework of an unstructured mesh multi-phasefluid model (Fluidity) and a combined finite-discrete element solid model (Y2D), there-fore, the governing equations, coupling methods and fracture modelling methods usedin those models are described in this section.

2.1. Governing equations for compressible fluids under the framework of ”Fluidity”

The governing equations for compressible fluids in Fluidityhave the followingform,

∂ρ

∂t+ ∇ · (ρu) = 0, (1)

∂

∂t(ρu) + ∇ · (ρuu − σ) = ρF, (2)

whereρ denotes the density,u is the velocity vector,t represents the time,σ is thestress tensor,F is the volume or internal force per unit mass (e.g., gravity). The densityρ is calculated by the equation of state:

P = ρ(γ − 1)e, (3)

whereP is the pressure,γ = Cp/Cv is a heat capacity ratio (Cv andCp being the specificheat at constant volume and at constant pressure respectively), e = CvT is the internalenergy per unit mass (“the specific internal energy”).

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2.2. Governing equations for solid dynamics

The fluid model, Fluidity, is coupled with Y2D, which resolves the solid dynamicsby a finite strain method [38]. The governing equation in solid mechanics is given by:

Fe + Fv + Fp + Fc = m∂us

∂t+ Fi , (4)

whereFe denotes the external force,Fv is the viscous force between the solid and fluid,Fp denotes the pressure between the fluid and solid,Fc is the contact force amongmultiple solids,m is the mass,us is the velocity of the solid andt is the time. Foradditional details, see [38].

2.3. Coupling methods between the fluid model (Fluidity) andsolid model (Y2D)

2.3.1. Coupling equationsA supplementary equation is introduced to couple the fluid code (Fluidity) and solid

code (Y2D), that is,ρ f

∆t(u f − u f

f ) =ρ f

∆t(us

s − usf ), (5)

where∆t is the time step and û f is the bulk velocity ( û f = α f uff + αsu

fs = u f

f + u fs(α f

andαs being the volume fractions of fluid and solid,α f + αs=1)). Subscripts denote thematerial fields, that is,s denotes the solid andf denotes the fluid. Superscripts denotethe mesh associated with the material (sdenotes values on the solid mesh andf denotesvalues on the fluid mesh). The solid velocity on the solid meshus

s is projected onto thefluid mesh in order to obtain the solutions of the coupled system, then it becomes û f

s

[39].The coupling process is achieved by introducing a source term sc into the momen-

tum equation (2), which represents the effects of forces of the solid on the fluid. Themomentum equation (2) then has the form of:

∂

∂t(ρu) + ∇ · (ρuu − σ) = ρF + sc. (6)

The source termsc considers an exchange of force between the solid and fluid, and hasthe formsc = (s f

c,x, sfc,y, s

fc,z)T . For additional details, see [40].

The continuity equation has the form of,

∇ · u f = 0, (7)

where

u f =

u ff if α f = 1, αs = 0

u fs if α f = 0, αs = 1.

(8)

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2.3.2. Coupling source termThe coupling process involves the calculation of the sourcetermsc in equation (6),

which is described briefly here (for additional details, see[40]). The viscosity forcesFs

viscosityandFspressureare calculated by:

Fsviscosity+ Fs

pressure=

∫

Γsolid

Ni n · (τsolid+ Ip)dΓ, (9)

whereN is the basis function,τsolid

is the viscous stress term;Γsolid is the solid surface,

n is the unit normal vector on the solid surfacen = (nx, ny, nz). I is the identity matrix.Once obtainingFs

viscosityandFspressure, the velocity of solidsus=(us, vs,ws) can be

calculated by equation (4). The source term can then be obtained using the followingequations:

s fc,x = axxus+ axyvs + axzws,

s fc,y = ayxus+ ayyvs + ayzws,

s fc,z = azxus+ azyvs+ azzws,

(10)

wherea denotes the viscosity coefficients and the subscriptx, y andz denote the co-ordinate directions,∆xwall is the fluid element length scale around the wall.∆r is thethickness of the shell, which is an intermediate thin area between the fluid and solid,and is introduced for calculating the impact of the solid on the fluid [39].

2.4. Fracture modelling

The fracture model in the combined finite-discrete element solid model (Y2D)treats the whole domain as a multi-body system. Each body is discretised into the finiteelement mesh. The fracture model is comprised of the the finite element formulationand discrete element formulation. The finite element formulation is used to model con-tinuum behaviour (i.e. calculation of stress and strain) before fractures are generated. Ifthe failure criterion is met, the discrete element formulation is then used for modelingdiscontinuum behaviour (contact forces between discrete bodies and distribution of thecontact force to nodes). The combination of the finite element formulation and discreteelement formulation ensures the transition from continuumbehaviour to discontinuumbehaviour can be captured accurately. The method is known asthe combined finite-discrete element method (FEMDEM) [41, 42]. A Mohr-Coulomb failure criterion witha tension cut-off is used.The overall fracture modelling algorithm based on FEMDEM isgiven in algorithm1(for details, see [41, 42]), where1, ut

solid denotes the solid velocity vector at each nodeat the time stept, uaccelerationis the acceleration,∆t is the time step,fexternaland finternal

are the external and internal forces at each node respectively, andm is the mass of thenode.

3. Model reduction

In reduced order modelling, any variable can be expressed asa linear combinationof a number of basis functions representing the original high fidelity modelling system

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Algorithm 1: Fracturing simulation(1) Input data.(2) Insert 6-node joint elements between 4-node tetrahedral elements.(3) Calculate stresses using the finite element formulation.(4) Judge whether new fractures are generated.if new fractures are generatedthen

add new contact forces.else

detect contact forces in DEM domain.end if(5) Calculate contact forces in DEM domain.(6) Calculate velocity of each node through explicit time integration.

ut+1solid = ut

solid + uacceleration∆t

uacceleration=fexternal− finternal

m(7) Output data.(8) Goto step (3): calculate stresses using the finite element formulation.(9) Stop.

in an optimal sense. It has the following form:

ϕ = ϕ +

m∑

i=1

αiΦi , (11)

whereϕ denotes a variable to be solved (e.g. the velocity, pressure, density and solidconcentration),ϕ is the mean of variable solutions over the simulation time period, αdenotes the POD coefficients,m is the number of POD bases andΦ denotes the PODbasis functions. Using POD, the basis functions can be calculated from snapshots ofvariable solutions recorded at regular time intervals. Theradial basis function inter-polation method is used to calculate the POD coefficients. The procedure of POD issummarized in algorithm2.

The radial basis function interpolation is used to determine the POD coefficients in(11). Commonly used RBFs are plate spline, multi-quadric, inverse multi-quadric andGaussian. RBFs have been widely used in the context of multidimensional interpola-tion. An interpolation functionf (x) representing a physical problem can be approxi-mated through a linear combination of the RBFφ centred atN points. In this work,the Gaussian RBF is used to construct the interpolation function f (x). The GaussianRBF has a form ofφ(r) = e−(r/σ)2

(r being the radius andσ being the shape parameter).In the following paragraph, a set of interpolation functions or hypersurfaces is derivedthrough POD-RBF method. The POD-RBF NIROM method was firstlypresented byXiao, et al. [35]. In this work this method is used to derive NIROM for the com-pressible fluid/solid problem and fracture problem. The form of the equations used for

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Algorithm 2: Proper Orthogonal Decomposition

(1) Compute solution of the coupled compressible fluid and solid system at timelevels 1, ...,Ns ;

(2) Retrieve snapshots matrix A from the solutions obtained;

(3) Subtract the mean of snapshots matrix A,i.e. A′ = A− Amean;

(4) Perform Singular Value Decomposition (SVD) to snapshots matrixA or A′, i.e.A = EΣFT ;

(5) Choose the dimension of ROM,m (m< Ns);

(6) Obtain POD basis functionsΦi = E:,i, for i ∈ {1, 2 . . .m} ;

solving the reduced system is:

αnu, j = fu, j(αn−1

u , αn−1p , α

n−1d , α

n−1c ) (12)

αnp, j = fp, j(αn−1

u , αn−1p , α

n−1d , α

n−1c ) (13)

αnd, j = fd, j(αn−1

u , αn−1p , α

n−1d , α

n−1c ) (14)

αnc, j = fc, j(αn−1

u , αn−1p , α

n−1d , α

n−1c ), (15)

whereα denotes POD coefficients, subscriptsu, p, d andc denote velocity, pressure,density and solid concentration components, subscriptj is the jth POD coefficient of acomplete set of POD coefficient (αu, αv, αd, αc), n is time step,f is a set of hypersur-face functions representing the reduced system.

The hypersurface functions are constructed using POD-RBF method, as describedin algorithm3, whereN denotes the number of data points.m denotes the number ofPOD basis functions.A is the matrix associated with the data point and centrec and

Ai, j = φ(∥

∥

∥

∥(αi, j

u , αi, jp , α

i, jd , α

i, jc ) − c

∥

∥

∥

∥).

7

Algorithm 3: Constructing a set of hypersurface using POD-RBF

(1) Generate a number of snapshots over the time period [0,T] by solving thecompressible fluid/solid interaction problem and fracture model;

(2) Calculate POD basis functionsΦu,Φp,Φd andΦc through a truncated SVD of thesnapshots matrix;

(3) Obtain the functional valuesyi at the data pointαtu, α

tp, α

tv via the solutions from

the high fidelity full model, wheret ∈ {1, 2, . . .T};

(4) Obtain a set of hypersurfaces through the following loop:

for j = 1 to mdo

(i) Calculate the weightswi, j by solving (16);

Awi, j = yi, j, i ∈ {1, 2, . . . ,N},

(ii) Obtain a set of hyper surfaces (fu, j , fp, j , fd, j , fc, j) by substituting the weightvalues obtained in the above step into equation (16);

fu, j(αu, αp, αd, αc) =N∑

i=1

wi, jφ j(∥

∥

∥(αu, αp, αd, αc) − (αiu, α

ip, α

id, α

ic)∥

∥

∥),

fp, j(αu, αp, αd, αc) =N∑

i=1

wi, jφ j(∥

∥


ip, α

id, α

ic)∥

∥

∥),

fd, j(αu, αp, αd, αc) =N∑

i=1

wi, jφ j(∥

∥


ip, α

id, α

ic)∥

∥

∥),

fv, j(αu, αp, αd, αc) =N∑

i=1

wi, jφ j(∥

∥


ip, α

id, α

ic)∥

∥

∥),

endfor

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Algorithm 4: Online NIROM calculation for compressible fluid and solid inter-action and fracture model

(1) Initialisation.for j = 1 to mdoInitialize α0

u, j , α0p, j , α

0d, j andα0

c, j ;

endfor

(2) Calculate solutions at current time step:for n = 1 to T dofor j = 1 to mdo

Solving fluid process:

(i) Evaluate the hypersurfacesf at the previous time stepn− 1 by using thecomplete set of POD coefficientsαn−1

u, j , αn−1p, j , αn−1

d, j andαn−1c, j :

fu, j ← (αn−1u , α

n−1v , α

n−1d , α

n−1c ), fp, j ← (αn−1

u , αn−1v , α

n−1d , α

n−1c ),

fd, j ← (αn−1u , α

n−1v , α

n−1d , α

n−1c ), fc, j ← (αn−1

u , αn−1v , α

n−1d , α

n−1c ),

(ii) Calculate the POD coefficientsαnu, αn

p, αnd andαn

c at the current time stepnusing the following equations:

αnu, j =

N∑

i=1

wi, jφi, j(r), αnp, j =

N∑

i=1

wi, jφi, j(r), (16)

αnd, j =

N∑

i=1

wi, jφi, j(r), αnc, j =

N∑

i=1

wi, jφi, j(r),

endfor

Calculate the solutionun, pn, dn andcn on the full space at current time stepn by projectingαn

u, j , αnp, j , α

nd, j andαn

c, j onto the full space.

un =

m∑

j=1

αnu, jΦu, j , pn =

m∑

j=1

αnp, jΦp, j , dn =

m∑

j=1

αnd, jΦd, j, cn =

m∑

j=1

αnc, jΦc, j,

Solving solid-fluid coupling:Fs

viscosity+ Fspressure=

∫

ΓsolidNin · (τ

solid+ Ip)dΓ,

obtains fc = (s f

c,x, sfc,y, s

fc,z)T using (10).

endfor

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4. Application to compressible fluid and solid problems

The FSI NIROM has been implemented under the framework of an advanced 3Dunstructured mesh multi-phase fluid model (Fluidity) and a combined finite-discreteelement solid model (Y2D). The compressible FSI NIROM is validated by a bendingbeam forced by flows first, then further validated by a more complex case: blasting.

4.1. Case 1: a bending beam forced by flows

The first case is a bending beam forced by flows. In this case, a solid beam isembedded in fluids with subject to a high pressure wave. The domain consists of arectangle of non-dimensional size of 4× 2 with 7500 nodes. The beam is located at thebottom center and has a size of 0.286× 1. The initial pressure at the area (0< x < 1.5)with a density of 8 is set as 116.5 Pa and the rest of the domain with a density of 1.5 isset as 1Pa. A slip boundary condition is applied on the left, bottom and the top sides.The momentum out boundary condition is applied on the right side. The density of thesolid is 100. The high fidelity full model was simulated during the time period [0, 0.8]with a time step size of∆t = 0.001. 800 snapshots were taken at a regularly spacedtime interval of 0.001.

4.1.1. Results with mean subtraction before performing PODThe FSI NIROM was first formed with mean subtraction before performing POD.

In this case, 30 POD basis functions were chosen to form the FSI NIROM. The singulareigenvalues associated to the chosen 30 POD bases are given in figure1. It can be seenthat 30 leading POD basis functions are capable of capturingalmost 99.5% of energyin the orginal dynamic system.

The pressure results from both the Fidelity model and FSI ROMare shown in figure2. It is illustrated that these FSI NIROM results are not good in comparison with thosefrom the high fidelity model. To further assess the quality ofthe FSI NIROM with meansubtraction before performing POD, the error analysis is carried out. The root meansquare error (RMSE) and correlation coefficient of results between the FSI NIROMand the fidelity model are shown in figure3. It can be seen that the accuracy of FSIROM results is low and need to be improved.

10

5 10 15 20 25 30NUMBER OF POD BASES

0

200

400

600

800

1000

SIN

GU

LAR

EIG

EN

VA

LUE

S

Figure 1: case 3: the figure shows the singular eigenvalues inorder of decreasing magnitude under thecondition that no mean is involved in SVD process.

(a) full model, t = 0.3 (b) full model, t = 0.8

(c) SVD with mean subtraction (30 POD bases),t = 0.3 (d) SVD with mean subtraction (30 POD bases),t = 0.8

Figure 2: Bending beam forced by flows case with mean subtraction: Pressure solutions comparison betweenthe full model and FSI NIROM with 30 POD bases at time instances t = 0.3 andt = 0.8.

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0 100 200 300 400 500 600 700Timestep

0

0.2

0.4

0.6

0.8

corr

elat

ion

coef

ficie

nt

SVD without mean, 30 POD bases

0 100 200 300 400 500 600 700Timesteps

20

22

24

26

28

30

32

34

36

38

40R

MS

E SVD without mean, 30 POD bases

(a) Correlation coefficient (b) RMSE

Figure 3: Bending beam forced by flows case with mean subtraction: The correlation coefficient and RMSEof pressure solutions between the high fidelity and FSI NIROMwith 30 POD bases.

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4.1.2. Results without mean subtraction before performingSVDIn this subsection, the FSI NIROM results without mean subtraction before per-

forming SVD are given and discussed. Figure4 presents the singular eigenvalues in adecreasing magnitude order. The figure4 shows that the first 12 singular eigenvaluesdecrease much more rapidly than other singular eigenvalues. The first 12 POD basesassociated with the first 12 singular eigenvalues have already captured 87.59% of thetotal energy.

Figure5 shows the pressure solutions comparison between the full model and FSINIROM with 12, 18 and 30 POD bases at time instancest = 0.3 andt = 0.8. We cansee that the results are much better than those with mean subtraction in figure2. Theresults with mean subtraction being in all probability averaged by mean. It also showsthat FSI NIROM using 12 POD bases performs well and the results of FSI NIROM areimproved by increasing the number of POD bases, especially at the front (see figure5(a), (c), (e), (g)). In order to see the effects of improvements by increasing number ofPOD bases, the absolute error of pressure between high fidelity model and FSI NIROMwith different number of POD bases at time instancest = 0.3 andt = 0.8 is given infigure6. The figure clearly shows that the error of the FSI NIROM relative to the highfidelity model become smaller as a larger number of POD bases is used.

In order to further validate the accuracy of the FSI NIROM without mean subtrac-tion, correlation and RMSE are used, see figure6. It is shown in this figure that the FSINIROM is very close to high fidelity model even when only 12 PODbases are usedand the error is decreased as the number of POD bases are increased.


0

500

1000

1500

SIN

GU

LAR

EIG

EN

VA

LUE

S

Figure 4: the figure shows the singular eigenvalues in order of decreasing magnitude without mean subtrac-tion before performing SVD.

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(c) SVD without mean subtraction (12 POD bases),t = 0.3 (d) SVD without mean subtraction (12 POD bases),t = 0.8

(e) SVD without mean subtraction (18 POD bases),t = 0.3 (f) SVD without mean subtraction (18 POD bases),t = 0.8

(g) SVD without mean subtraction (30 POD bases),t = 0.3 (h) SVD without mean subtraction (30 POD bases),t = 0.8

Figure 5: Bending beam forced by flows case without mean subtraction: Pressure solutions comparisonbetween the full model and FSI NIROM with 12, 18 and 30 POD bases at time instancest = 0.3 andt = 0.8.

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(a) error (12 POD bases),t = 0.3 (b) error (12 POD bases),t = 0.8

(c) error (18 POD bases),t = 0.3 (d) error (18 POD bases),t = 0.8

(e) error(30 POD bases),t = 0.3 (f) error(30 POD bases),t = 0.8

Figure 6: Bending beam forced by flows case without mean subtraction: Error between the high fidelitymodel and FSI NIROM with 12, 18 and 30 POD bases at time instancest = 0.3 andt = 0.8.

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0 100 200 300 400 500 600 700Timestep

0.9515

0.957

0.9625

0.968

0.9735

0.979

0.9845

0.99

0.9955

1.001

corr

elat

ion

coef

ficie

nt

12 POD bases18 POD bases30 POD bases

0 100 200 300 400 500 600 700Timesteps

0

2

4

6

8

10

RM

SE 12 POD bases

18 POD bases30 POD bases


Figure 7: Bending beam forced by flows case without mean subtraction: RMSE and correlation coefficientof pressure solutions between the high fidelity and FSI NIROMwith 12, 18 and 30 POD bases.

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4.2. Blasting test case

In the second example a blasting-induced fracture test caseis resolved. The com-putational domain is presented in figure15 which shows a solid square block with asize of 2× 2 m embedded within a compressible gas rectangle area with a size of 3× 3m. The explosion point lies at the center of the computational domain with a diameterof 0.2m and a very high initial pressure. The initial high pressure of the explosion pointis set to be 108 Pa and the initial high temperature is set to be 1000 Kelvin. The back-ground area (excluding the explosion point) has an initial pressure of 101325 Pa andan initial temperature of 273.26 Kelvin. The viscosityµ is 0.1 Pa· s. The solid with adensity of 2340kg/m3 has a penalty number of 2.0×1010 and a Youngs modulus E of2.66×1010. The tensile strength and the shear strength are 4×106 Pa and 1.4×107 Parespectively. The energy decrease rate is 200.

The high fidelity full model was simulated with a finite element mesh of 48600nodes during the time period [0, 0.2] with a time step size of∆t = 0.00008. 250snapshots were taken at a regularly spaced time intervals of∆t = 0.0008.

4.2.1. Results with mean subtraction before performing SVDIn this section, results from NIROM with mean subtraction before performing the

SVD are presented. Figure8 presents the velocity solutions from the high fidelity fullmodel and FSI NIROM with 100 POD bases at time instancest = 0.04 andt = 0.16.It is shown that the structure of flows obtained from the FSI NIROM is similar to thatfrom the high fidelity model, but there are some large errors in velocity values. Figure9 shows the pressure solutions from the high fidelity full model and FSI NIROM with100 POD bases at time instancest = 0.04 andt = 0.16. It again shows that the resultsfrom FSI NIROM need to be improved.

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(c) (100 POD bases),t = 0.04 (d) (100 POD bases),t = 0.16

Figure 8: Blasting case with mean subtraction: Velocity solutions comparison between the full model andFSI NIROM with 100 POD bases at time instancest = 0.04 andt = 0.16.

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Figure 9: Blasting case with mean subtraction: Pressure solutions comparison between the full model andFSI NIROM with 100 POD bases at time instancest = 0.04 andt = 0.16.

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4.2.2. solutions without mean subtraction before performing SVDIn this section, solutions without mean subtraction beforeperforming SVD are

given. Figure10 shows the singular eigenvalues in order of decreasing magnitudewithout mean subtraction before performing SVD. In this example the rate of decreasein the singular values is fast, the fifteenth singular value is considerably smaller thanthe first singular value. According to the decreasing rate ofthe singular values− asshown in figure10, 6, 12 and 50 POD basis functions were chosen to illustrate thecapability of the FSI NIROM.

Figure11shows the velocity solutions comparison between the full model and FSINIROM with 6, 12 and 50 POD bases at time instancest = 0.04 andt = 0.16. Itis evident that the FSI NIROM with only 6 POD basis functions perform well whenmean was not subtracted before SVD, even better than solutions from FSI NIROMwith 100 POD basis functions when the mean was subtracted before SVD− as shownin figure 8. For velocity solutions, there is no obvious difference between the highfidelity model and FSI NIROMs. The figure11also shows that shock front of the blastwave is captured very well by increasing the number of POD bases from 6 to 50. Thereis no visible difference between the high fidelity model and FSI NIROM with 50 PODbases. In order to see clearly the effects of choosing larger number of POD bases, theerrors between the high fidelity model and FSI NIROM with 6, 12and 50 POD basisfunctions at time instancest = 0.04 andt = 0.16 are presented in figure12. It is evidentthat greater accuracy is obtained by choosing larger numberof POD bases.

Figure13 presents the pressure solutions comparison between the full model andFSI NIROM with 6, 12 and 50 POD bases at time instancest = 0.04 andt = 0.16. Thepressure solutions from FSI NIROM using 6 POD bases are not asgood as velocitysolutions from FSI NIROM using the same number of POD bases. In this case, thereare visible differences between the high fidelity model and FSI NIROM using 6 and12 POD bases, which is evident at the time instancet = 0.16. The errors between thehigh fidelity model and FSI NIROM with 6, 12 and 50 POD basis functions at timeinstancest = 0.04 andt = 0.16 are plotted in figure14. It is evident that the error isdecreased by choosing more POD basis functions.

In order to assess the performance of the FSI NIROM, velocitysolutions obtainedfrom the high fidelity model and FSI NIROMs at a point (x = 1.5, y = 1.6333) nearthe explosion point over the simulation time period is plotted in figure15. One reasonthat we choose the point around the explosion centre is that there is an abrupt changearound the explosion point. The figure15illustrates that FSI NIROM with less numberof POD basis functions perform well when there is no abrupt changes, whereas FSINIROM with 50 POD basis functions captures the abrupt changes very well.

The accuracy of the FSI NIROM is validated by RMSE and correlation coefficients,which is shown in figure16. It is shown that the RMSE of pressure results decreasesas the number of POD bases increases. The correlation coefficients from FSI NIROMsare over 0.935, indicating that the high fidelity model and FSI NIROMs are highlycorrelated. The FSI NIROM gets more closer agreement to highfidelity model as thenumber of POD basis functions increases.

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0

500

1000

1500

2000

2500

3000

SIN

GU

LAR

EIG

EN

VA

LUE

S

Figure 10: Blasting case: The singular eigenvalues in orderof decreasing magnitude without mean subtrac-tion before performing SVD.

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(e) (12 POD bases),t = 0.04 (f) (12 POD bases),t = 0.16

(g) (50 POD bases),t = 0.04 (h) (50 POD bases),t = 0.16

Figure 11: Blasting case without mean subtraction: Velocity solutions comparison between the full modeland FSI NIROM with 6, 12 and 50 POD bases at time instancest = 0.04 andt = 0.16.

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Figure 12: Blasting case without mean subtraction: Velocity error between the high fidelity model and FSINIROM with 6, 12 and 50 POD bases at time instancest = 0.04 andt = 0.16.

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(e) (12 POD bases),t = 0.04 (f) (12 POD bases),t = 0.16

(g) (50 POD bases),t = 0.04 (h) (50 POD bases),t = 0.16

Figure 13: Blasting case without mean subtraction: pressure solutions comparison between the full modeland FSI NIROM with 6, 12 and 50 POD bases at time instancest = 0.04 andt = 0.16.

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Figure 14: Blasting case without mean subtraction: Pressure error between the high fidelity model and FSINIROM with 6, 12 and 50 POD bases at time instancest = 0.04 andt = 0.16.

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0 50 100 150 200 250

0

10

20

Timestep

Velocity

full model

6 POD bases

12 POD bases

50 POD bases

Figure 15: Blasting case: Velocity comparison at a point (x = 1.5, y = 1.6333).

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0 50 100 150 200 250Timestep

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

corr

elat

ion

coef

ficie

nt

6 POD bases12 POD bases50 POD bases

0 50 100 150 200 250Timesteps

0

2e+05

4e+05

6e+05

8e+05R

MS

E 6 POD bases12 POD bases50 POD bases


Figure 16: blasting case without mean subtraction: The correlation coefficient and RMSE of pressure solu-tions between the high fidelity and FSI NIROM with 6, 12 and 50 POD bases.

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4.3. Efficiency of the NIROM

In this section, we compare the computational cost requiredfor running the highfidelity full model with the online NIROM computation involved in algorithm4. Thesimulations were carried out on a 12 cores (Intel(R) Xeon(R)X5680) workstation with48GB RAM. During the simulations, only one core with 3.3GHz is used. The processof constructing a set of hypersurfaces involved in algorithm 3 is precomputed, there-fore, it is not listed in the table. As shown in table1, the computational time requiredfor running the NIROM is decreased drastically. In blastingtest case with 48600 nodes,the NIROM obtained a CPU speed-up of 5 orders of magnitude.

Table 1: Comparison of the online CPU cost (dimensionless) required for running the high fidelity modeland NIROM during one time step.

Cases Model Assembling and Projection Interpolation TotalSolving

Bending Full model 4.95120 0 0 4.95120beam NIROM 0 0.0003 0.0001 0.00040

Full model 224.47059 0 0 224.47059Blasting NIROM 0 0.0003 0.0001 0.00040

5. Conclusion

A POD-RBF NIROM has been, for the first time, applied to a structure interactingwith compressible fluid flows and fracture models and implemented under the frame-work of a combined finite-discrete element method based solid model (Y2D) and anunstructured mesh finite element model (Fluidity). The model is independent of thegoverning equations, therefore, it is easy to modify and implement. The performance ofthe NIROM for compressible fluid/solid interactions and fracture models is numericallyillustrated for two test cases: a bending beam forced by flowsand a blasting case. Themean subtraction issue before performing POD is discussed by comparison betweensolutions with and without subtraction. An error analysis was carried out to validateand assess the newly NIROM. The numerical results show that the NIROM performswell and exhibits a good agreement with high fidelity model. The front around thebeam is captured well using only a few number of POD bases without mean subtrac-tion beforehand (see figure5). The computational cost of the NIROM is comparedagainst high fidelity full model. The CPU cost required for NIROM can be reducedby a factor of several orders of magnitude. In addition, the CPU cost is independentof number of nodes on computational mesh which is a particular important factor af-fecting simulation time of high fidelity full model. Future work includes extending thismodel to problems with variable material properties.

Acknowledgments

This work was carried out under funding from Janet Watson scholarship at Departmentof Earth Science and Engineering. Authors would like to acknowledge the support

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of the UK’s Natural Environment Research Council projects(NER/A/S/2003/00595,NE/C52101X/1 and NE/C51829X/1), the Engineering and Physical Sciences ResearchCouncil (GR/R60898, EP/I00405X/1 and EP/J002011/1), and the Imperial CollegeHigh Performance Computing Service. Prof. I.M. Navon acknowledges the supportof NSF/CMG grant ATM-0931198. Dr. Xiao acknowledges the support ofNSFC grant11502241 and China postdoctoral science foundation grant (2014M562087). Dr. Chenand Dr. Xiang acknowledges the support of NSFC grant 51479146. Prof. Pain and Dr.Fang are grateful for the support provided by BP Exploration. Prof. Pain is grateful forthe support of the EPSRC MEMPHIS multi-phase flow programme grant. The authorsacknowledge C. E. Heaney for comments that helped improve the manuscript.

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